Title Author(s) Citation ON THE CLASSIFICATION OF HOMOGENEOUS -SPHERES IN COMPLEX GRASSMANNIANS Fei, Jie; Jiao, Xiaoxiang; Xiao, Liang; Xu, Xiaowei Osaka Journal of Mathematics 5(1) P135-P15 Issue Date 13-3 Text Version publisher URL https://doiorg/11891/4475 DOI 11891/4475 rights
Fei, J, Jiao, X, Xiao, L and Xu, X Osaka J Math 5 (13), 135 15 ON THE CLASSIFICATION OF HOMOGENEOUS -SPHERES IN COMPLEX GRASSMANNIANS JIE FEI, XIAOXIANG JIAO, LIANG XIAO and XIAOWEI XU (Received November 6, 9, revised May 3, 11) Abstract In this paper we discuss a classification problem of homogeneous -spheres in the complex Grassmann manifold G(k 1, n 1) by theory of unitary representations of the 3-dimensional special unitary group SU() First we observe that if an immersion x Ï S G(k 1, n 1) is homogeneous its image x(s ) is a -dimensional (SU())-orbit in G(k 1, n 1), where Ï SU() U(n 1) is a unitary representation of SU() Then we give a classification theorem of homogeneous -spheres in G(k 1, n 1) As an application we describe explicitly all homogeneous -spheres in G(, 4) Also we mention about an example of non-homogeneous holomorphic -sphere with constant curvature in G(, 4) 1 Introduction It is one of fundamental problems in differential geometry to study the rigidity and homogeneity of special surfaces and submanifolds in a given Riemannian manifold The most important and interesting case is that the ambient manifold is a space form For example, minimal surfaces with constant (Gaussian) curvature in real space forms have been classified completely ([3], [4], [11]), and minimal -spheres with constant curvature in the complex projective space P n also have been classified completely ([1], []) We wish to study -spheres with constant curvature immersed in the complex Grassmannians which is a generalization of the complex projective space P n The complex Grassmann manifold G(k1,n1) is the set of all (k1)-dimensional complex vector subspaces in n1, which is isomorphic to a Hermitian symmetric space U(n 1)(U(k 1) U(n k)) We equip G(k 1,n 1) with a canonical Kähler metric which is U(n1)-invariant and has Einstein constant (n1) Particularly, G(1,n1) is the complex projective space P n, which has constant holomorphic sectional curvature 4 However, the geometric structure of G(k 1, n 1) is much more complicated when k 1 For example, when k 1, G(k 1, n 1) does not have constant holomorphic sectional curvature, and the rigidity of holomorphic curves in G(k 1, n 1) fails ([7]) For this reason, it is hard to generalize some perfect results of submanifolds in P n 1 Mathematics Subject Classification 53C4, 53C55 This project was supported by the NSFC (Grant No 117148, No 117149, No 1111389) and the Knowledge Innovation Program of the Chinese Academy of Sciences (KJCX3-SYW-S3) The fourth author was supported by the Fundamental Research Funds for the Central Universities
136 J FEI, X JIAO, L XIAO AND X XU to the ones of submanifolds in a general complex Grassmannians However, when the integers k, n are small, there are some results about minimal -spheres in G(k 1,n1) Minimal -spheres with constant curvature in G(, 4) were determined by Z-Q Li and Z-H Yu ([14]), and holomorphic -spheres with constant curvature in G(, 5) were also investigated by X-X Jiao and J-G Peng ([1]) In [1], S Bando and Y Ohnita proved that all minimal -spheres with constant curvature in P n are homogeneous, and later, H-Z Li, C-P Wang and F-E Wu classified homogeneous -spheres in P n in [13] by the method of harmonic sequence The purpose of this paper is to classify homogeneous -spheres in complex Grassmannians (see Theorem 41), which is a generalization of [13] We should point out that our original idea derives from [1] and the observations of the main result in [13] They inspire us to consider this problem from the angle of the unitary representation theory of SU(), which is quite different from the method used in [13] As its applications we give an explicit description of all homogeneous -spheres in G(, 4) and their differential geometric quantities (see Theorems 51 and 5) It is worthy to notice that minimal -spheres with constant curvature in spheres ([8], [9]) and complex projective spaces ([1]) are always homogeneous, but it is not true for the case when the ambient space is a general complex Grassmannians To show this phenomenon, we prove the non-homogeneity for an example of holomorphic -sphere with constant curvature in G(, 4) given in [14] (see Theorem 53) Therefore, this phenomenon also reflects the complexity of the geometric structure of general complex Grassmannians Our paper is organized as follows In Section, we recall some basic facts of unitary representations of SU() In Section 3, we prove that if an immersion x Ï S G(k 1, n 1) is homogeneous its image x(s ) is a -dimensional (SU())-orbit in G(k 1, n 1), where Ï SU() U(n 1) is a unitary representation of SU() In Section 4, we give a classification theorem of homogeneous -spheres in G(k 1,n1) which generalizes main theorem of H-Z Li, C-P Wang and F-E Wu in the case of P n In Section 5, we describe explicitly all homogeneous -spheres in G(, 4) Preliminaries In this section, we will agree on the same notations in [1], and begin with recalling some basic facts of irreducible unitary representations of the 3-dimensional special unitary group SU(), which is defined by a b Æ SU() g and its Lie algebra su() is given by su() X b Æa Á a, b ¾, a b 1 1x Æy y 1x Á x ¾ Ê, y ¾,,
HOMOGENEOUS -SPHERES IN COMPLEX GRASSMANNIANS 137 with a natural basis { 1,, 3 }, which is given by 1 1 1, 1 1, 3 1 1 Set T {exp( 1 )Á ¾ Ê} we have the homeomorphism S ³ SU()T The tangent space of the point [T ] ¾ S can be naturally identified with the subspace span Ê {, 3 } of su() We define a complex structure on S such that 1 3 is a vector of (1, )-type For each nonnegative integer n, let V n be the representation space of SU(), which is an (n 1)-dimensional complex vector space of all complex homogeneous polynomials of degree n in two variables z and z 1 The standard irreducible representation n of SU() on V n is defined by (1) n (g) f (z, z 1 ) Ï f (az bz 1, Æ bz Æaz 1 ), where g ¾ SU() and f ¾ V n If we view the elements of V n as polynomial functions of S 3 {(z, z 1 ) ¾ Á z z 1 1}, we can define a SU()-invariant Hermitian inner product (, ) on V n as follows (n 1)! ( f, h) Ï S 3 f Æ h dú, where h ¾ V n defined by and dú is the volume element of S 3 It is easy to check that {u (n) k } u (n) k Ï 1 k! (n k)! z n k z k 1, k n, is a unitary basis of V n Since n (g)u (n) k n (g)u (n) k ¾ V n, we can write n l l k (a, b)u(n) l, where { l k (a, b)} are polynomials of degree n in {a, Æa, b, b} Æ By (1), we have () l k (a, b) l! (n l)! n k k a p (Æa) k q b n k p ( b) Æ q k! (n k)! p q pqn l Let n1 be the complex number space of dimension (n 1), and {E i } i n be the standard basis of n1 With respect to the unitary basis {u (n) k } of V n, there is a natural isomorphism between V n and n1 Under such isomorphism, each linear endomorphism
à Éà 138 J FEI, X JIAO, L XIAO AND X XU n (g) (for g ¾ SU()) can be represented by a matrix ( l k (a, b)), which is still denoted by n (g) It is easy to see n (g) ¾ U(n 1), thus we have a Lie group homomorphism n Ï SU() U(n 1), g n (g) ( l k (a, b)) The representation n of SU() induces an action of su() on V n which can be described as follows (3) n (X)(u (n) k ) d (exp t X)(u (n) dt k ) t k(n k 1) Æyu (n) k 1 (n k) 1xu (n) k (n k)(k 1)yu (n) k1, for k n and X ¾ su() Using the matrix notation, we get a Lie algebra homomorphism n Ï su() u(n 1), X n (X), which is the differential of the homomorphism n From (3), in terms of matrix form, n (X) can be written as (4) n (X) n 1x ny n Æy (n ) 1x (n 1)y n Æy n 1x 3 Homogeneous -spheres in complex Grassmannians In this section, we shall reduce the classification problem of homogeneous -spheres in complex Grassmannians to an algebraic problem on unitary representations of SU() whose classification theory is classical and well-known An immersion x Ï S G(k 1, n 1) is said to be homogeneous, if for any two points p, q ¾ S there exists an isometry É of S and a holomorphic isometry of G(k 1, n 1) such that É (p) q and the following diagram communicates (31) S x à G(k 1, n 1) S x à G(k 1, n 1), ie, x Æ É Æx We can identify É (resp ) with an element of SO(3) (resp U(n1)) All such form a subgroup G of U(n 1) and G acts transitively on x(s ) It s known that such -spheres in G(k 1, n 1) have constant curvature, but they are non-minimal
HOMOGENEOUS -SPHERES IN COMPLEX GRASSMANNIANS 139 in general Standard examples of homogeneous -spheres in P n are so-called Veronese sequence which are completely determined by S Bando and Y Ohnita ([1]) As mentioned above, an immersion x Ï S G(k 1, n 1) is said to be homogeneous if the group G I (G(k 1, n 1), x(s )) acts transitively on x(s ), or equivalently, x(s ) is a -dimensional G-orbit in G(k 1, n 1) Since S is compact, we know that G is also a compact Lie subgroup of U(n 1) Lemma 31 The group Ç G {É ¾ SO(3)Á x Æ É x} consists the unit element I 3 only Proof Let É ¾ G, Ç then 1 is an eigenvalue of É, so the action of É on S has a fixed point p The differential map of x and É at p satisfy x p Æ É p x p Since x p is injective, it follows that É p is an identity map Thus, we obtain É I 3, which completes the proof By this lemma and the commutation diagram (31), we obtain a natural Lie group homomorphism Ï G SO(3), É Up to now, we can prove our following principle result Theorem 3 If xï S G(k 1,n 1) is a homogeneous immersion there exists a unitary representation Ï SU() U(n 1) such that x(s ) is a -dimensional (SU())-orbit in G(k 1, n 1) Proof Let g be the Lie algebra of G and Ï g o(3) be the Lie algebra homomorphism induced by Set H ker and h ker H is a closed normal Lie subgroup of G whose Lie algebra is h Obviously, h is an idea of g Since G is compact, one can equip g with an Ad G -invariant inner product, The orthogonal complement subspace of h in g with respect to the Ad G -invariant inner product is denoted by h Then h is a subalgebra of g and also an ideal of g So there exists a unique connected Lie subgroup K of G with its Lie algebra h By the fundamental homomorphism theorem of Lie algebra, we obtain the following Lie algebra isomorphisms: (h ) (g) gh h Since x(s ) is a -dimensional G-orbit and H acts on x(s ) keeping every point fixed, we know that x(s ) is also a -dimensional K -orbit Therefore, we obtain the relationship dim (h ) dimh dimk One can conclude that (h ) o(3) by the well known fact that there is no -dimensional subalgebra of o(3), and hence h o(3) Then we get a covering homomorphism K Ï K SO(3), which implies that K is
14 J FEI, X JIAO, L XIAO AND X XU isomorphic to SU() or SO(3) If K SU() Ï SU() K i U(n 1) defines a unitary representation of SU() If K SO(3), let be the adjoint representation of SU() the map Ï SU() SO(3) K i U(n 1) also defines a unitary representation of SU() This completes the proof Two homogeneous immersions xï S G(k 1,n1) and x Ï S G(k 1,n1) are said to be equivalent or congruent, if there exists an element A ¾ U(n 1) such that x A Æ x According to Theorem 3, the classification of equivalent classes of homogeneous -spheres in G(k 1, n 1) is reduced to the following two problems: (I) Classifying the equivalence classes of unitary representations of SU(); (II) Determining all (SU())-orbits in G(k1,n1) of dimension, here Ï SU() U(n 1) is a unitary representation of SU() The problem (I) is classical and well-known in the representation theory of compact Lie groups, namely, the set {( n, V n ), n, 1,, } forms all inequivalent irreducible unitary representations of SU() and every unitary representation of SU() can be expressed as direct sum of irreducible ones To solve the problem (II), we can prove the following theorem: Theorem 33 An orbit M of (SU()) on G(k 1, n 1) is a -dimensional sphere immersed in G(k 1, n 1) if and only if M goes through the point W ¾ G(k 1, n 1) which is a (k 1)-dimensional vector subspace invariant by T Proof Suppose that M W (SU()) is a -dimensional orbit for some W ¾ G(k 1, n 1), and H is the isotropy subgroup of SU() at the point W It is easy to see that H is a 1-dimensional closed Lie subgroup of SU() and its Lie algebra h is a 1-dimensional subalgebra of su() Thus there is an element X of su() such that h span Ê {X} According to some basic theory of linear algebra, there exist g ¾ SU() and a nonzero real number x such that g 1 Xg x 1 Hence g 1 hg span Ê { 1 } and it follows that g 1 H o g T, where H o is the connected component of H which contains the unit Taking the point W W (g) the isotropy group at W is g 1 Hg which contains T, ie, W is a (k 1)-dimensional vector subspace invariant by the action of T Thus, we prove the sufficiency of our theorem, and the necessity follows from the fact that su() has no -dimensional subalgebra 4 Classification theorem of homogeneous -spheres in G(k 1, n 1) In this section, we will give a classification theorem of homogeneous -spheres in G(k 1, n 1)
HOMOGENEOUS -SPHERES IN COMPLEX GRASSMANNIANS 141 In virtue of Theorem 33, the problem (II) was reduced to determining of all (k1)- dimensional vector subspaces invariant by T Let Ï SU() U(n 1) be a unitary representation of SU() and T Ï T U(n 1) be the restriction of from SU() to T Since T is a torus group, we just need to determine all 1-dimensional vector subspaces invariant by T If is irreducible, ie, n for some nonnegative integer n By (), it is easy to see that (41) e 1 e 1 n T Ï T U(n 1), diag { e 1n, e 1(n ),, e 1n } Hence, {E i, i n} are eigenvectors of n (T ) which belong to eigenvalues e 1n,, e 1(n i),, e 1n, respectively Thus, W i span {E i } Ï [E i ], i n, are all 1-dimensional vector subspaces invariant by T Define the map { (n) i } by (n) i Ï S SU()T P n, gt W i n (g) [ f (n) i ], which are SU()-equivariant immersions of S into P n, here f (n) i ( i, 1 i,, n i ) The sequence { (n), (n) 1,, (n) n } is well-known as Veronese sequence in Pn ([1], are []) The Gaussian curvature K and the Kähler angle of (n) i K 4 n i(n i), cos n i n i(n i), respectively If is reducible n1 nr n n 1 n r r 1, ie, and n1 n 11 n r 1 with Ï SU() U(n 1), g (g) diag{ n1 (g), n (g),, nr (g)} Set E n «j «Ï E i, where i n 1 n «1 j ««1 and j «n «It follows from (41) that a 1-dimensional vector subspace invariant by T must be spanned by a complex vector Ú with the following form (4) Ú c 1 E n 1 j 1 c r E n r j r, c «¾, 1 «r,
14 J FEI, X JIAO, L XIAO AND X XU where {n «, j ««1,, r} are nonnegative integers satisfying (43) n 1 j 1 n j n r j r In general, a (k 1)-dimensional vector subspace invariant by T can be spanned by k 1 complex vectors {Ú i i 1,, k 1}, where each Ú i has the form (4) and they satisfy (Ú i, Ú j ) Æ i j with respect to the standard Hermitian inner product (, ) on n1 Combining Theorems 3 and 33 together with the above arguments, we obtain the following classification Theorem 41 Let xï S G(k 1, n 1) be a homogeneous -sphere in G(k 1, n 1) Then there exist nonnegative integers {n «, j 1,««1,, r},, {n «, j k1,««1,, r} satisfying n 1 n r r n 1, j 1,«,, j k1,«n «(«1,, r), n 1 j 1,1 n j 1, n r j 1,r, n 1 j k1,1 n j k1, n r j k1,r and complex constants {c i,«i 1,, k 1, «1,, r} satisfying r «1 c i,«c h,«æ ji,«j h,«æ ih such that x A Æ f, where A ¾ U(n 1) and f is defined by f Ï S SU()T G(k 1, n 1), gt ¾ c 1,1 f (n 1) j 1,1 c 1, f (n ) j 1, c 1,r f (n r ) j 1,r c,1 f (n 1) j,1 c, f (n ) j, c,r f (n r ) j,r c k1,1 f (n 1) j k1,1 c k1, f (n ) j k1, c k1,r f (n r ) j k1,r REMARK Theorem 41 is a generalization of main theorem of H-Z Li, C-P Wang and F-E Wu in the case of P n ([13]) However in order to classify completely it is necessary to determine all {n «, j 1,««1,, r},, {n «, j k1,««1,, r} and {c i,«i 1,, k 1, «1,, r} satisfying the above conditions In next section, we will do it completely in the case of G(, 4) In the case of more general complex Grassmannians one would need more efforts to do them certainly
HOMOGENEOUS -SPHERES IN COMPLEX GRASSMANNIANS 143 5 Explicit description of homogeneous -spheres in G(, 4) In this section, we will describe explicitly all homogeneous -spheres in G(, 4), from which, one can find a family of non-minimal homogeneous -spheres in G(, 4) To do this well, we should consider the following four cases respectively CASE I If 3 Ï SU() U(4) and Ï su() u(4) can be given by () and (4) explicitly as follows (51) (5) (g) (X) a 3 3a b 3ab b 3 3a b Æ a(a b ) b(a b ) 3Æab 3ab Æ b(b Æ a ) Æa(a b ) 3Æa b Æb 3 3Æa b Æ 3Æa b Æ Æa 3 3 1x 3y 3 Æy 1x y Æy 1x 3y 3 Æy 3 1x By the arguments in Section 4, we know that [E k ], k 3 are all 1-dimensional vector subspaces invariant by T Then span {E k, E l }, k l 3 are all -dimensional vector subspaces invariant by T, so we get the following six homogeneous -spheres (I1) The base point W f Ï S G(, 4), gt (I1 ) The base point W f Ï S G(, 4), (I) The base point W f Ï S G(, 4), (I ) The base point W f Ï S G(, 4), 1 1, a 3 3a b 3ab b 3 3a b Æ a(a b ) b(a b ) 3Æab 1 1 3ab Æ b(b Æ a gt ) Æa(a b ) 3Æa b Æb 3 3Æa b Æ 3Æa b Æ Æa 3 1 1 gt a 3 3a b 3ab b 3 3ab Æ b(b Æ a ) Æa(a b ) 3Æa b 1 1 3a b Æ a(a b gt ) b(a b ) 3Æab Æb 3 3Æa b Æ 3Æa b Æ Æa 3
144 J FEI, X JIAO, L XIAO AND X XU (I3) The base point W 1 1 a f Ï S 3 3a G(, 4), gt b 3ab b 3 Æb 3 3Æa b Æ 3Æa b Æ Æa 3 (I3 ) The base point W f Ï S G(, 4), 1 1 3a b Æ a(a b gt ) b(a b ) 3ab Æ b(b Æ a ) Æa(a b ) 3Æab 3Æa b It is clear that the cases (Ii) and (Ii ) (i 1,, 3) are Hermitian orthogonal with respect to the standard Hermitian inner product of 4 CASE II If 1 Ï SU() U(4) and Ï su() u(4) can be written explicitly as follows (53) (54) (g) (X) a b Æb Æa 1 1, 1x y Æy 1x, by () and (4) Then, the restriction representation T Ï T U(n 1) is given by diag{e 1, e 1 } diag{e 1, e 1, 1, 1} So, we get two inequivalent homogeneous -spheres up to U(4)-equivalent (II1) The base point W 1 1 (II1 ) The base point W f Ï S G(, 4), gt 1 1 f Ï S G(, 4), gt a b 1 b Æ Æa 1 We know that (II1) and (II1 ) are also Hermitian orthogonal with respect to the standard Hermitian inner product of 4
HOMOGENEOUS -SPHERES IN COMPLEX GRASSMANNIANS 145 CASE III If, similarly, Ï SU() U(4) and Ï su() u(4) can be written as follows (55) (56) (g) (X) a ab b ab Æ a b Æab Æb Æa b Æ Æa 1 1x, y Æy y Æy 1x, by () and (4) The restriction representation T Ï T U(n 1) is given by diag{e 1, e 1 } diag{e 1, 1, e 1, 1} Hence, all 1-dimensional vector subspaces invariant by T are [E ], [E ] and [Ú], where Ú c 1 E 1 c E 3 with c 1 c 1 Then up to U(4)-equivalent, we have two isolate homogeneous -spheres and two 1-parameter families of homogeneous -spheres (III1) The base point W 1 1 f Ï S G(, 4), (III1 ) The base point W f Ï S G(, 4), (III) The base point W t f t Ï S G(, 4), gt (III ) The base point W t Éf t Ï S G(, 4), gt a gt 1 1 gt 1 cos t sin t ab b Æb Æa Æ b Æa ab Æ a b Æab 1, t ¾ [, ] a ab b ab Æ cos t (a b ) cos t Æab cos t sin t 1, t ¾ [, ] sin t cos t Æ b Æa Æ b Æa a Æ b sin t (b a ) sin t Æab sin t cos t It is easy to check that (IIIi) and (IIIi ) (i 1, ) are also Hermitian orthogonal with respect to the standard Hermitian inner product of 4
146 J FEI, X JIAO, L XIAO AND X XU CASE IV If 1 1 Ï SU() U(4) and Ï su() u(4) can be written as follows (57) (58) (g) (X) a b Æb Æa a b b Æ Æa, 1x y Æy 1x 1x y Æy, 1x by () and (4) The restriction representation T Ï T U(n 1) is given by diag{e 1, e 1 } diag{e 1, e 1, e 1, e 1 } Hence, the 1-dimensional vector subspaces invariant by T are [Ú 1 ] and [Ú ], where Ú 1 c 1 E c E and Ú d 1 E 1 d E 3 with c 1 c d 1 d 1 Then up to U(4)-equivalent, we have two isolate homogeneous -spheres and a family of homogeneous -spheres (IV1) The base point W (IV1 ) The base point W 1 1 f Ï S G(, 4), gt 1 1 f Ï S G(, 4), gt a b a b b Æ Æa b Æ Æa And also, (IV1) and (IV1 ) are Hermitian orthogonal with respect to the standard Hermitian inner product of 4 c1 c (IV) The base point W, with c 1 c d 1 d 1 d 1 d and Ï c 1 d c d 1 f (c 1, c, d 1, d )Ï S G(, 4), gt c1 a c 1 b c a c b d 1b Æ d1 Æa d b Æ d Æa Thus, we have completely classified homogeneous -spheres in G(, 4) Next, we will give some geometrical descriptions of these homogeneous -spheres in G(, 4) We only compute the geometric quantities of the case (I1) For other cases, we omit the details of calculations and just list the results in Table 1
HOMOGENEOUS -SPHERES IN COMPLEX GRASSMANNIANS 147 Let Đ (Đ A Æ B ), A, B 3 be the u(4)-valued right-invariant Maurer Cartan form of U(4) The Maurer Cartan structure equations of U(4) are (59) dđ A Æ B Đ A Æ C Đ C Æ B Then the canonical Kähler metric of G(, 4) and its Kähler form can be written as ds «,i Đ «Æ i Æ Đ«Æ i, ¾ 1 «,i Đ «Æ i Æ Đ«Æ i, where the range of the indices are «, 1 and i, 3 respectively We choose a unitary frame field e (e, e 1, e, e 3 ) along f, where e A E A (g), A, 1,, 3 It is easily see from (5) that the pull back of Maurer Cartan form can be written as (51) e Đ Æ 3 3 Æ 11 Æ 3 Æ 3 Æ 33 Æ with Æ 3 Æ 3, 1 Æ 1 Æ and Æ 3 1 Æ 1, where is a complex-valued (1, ) form of S, which defined up to a factor of absolute value 1, and the induced metric is f ds Æ If we write (511) f Đ A Æ B A Æ B a A Æ B b A Æ B Æ, and A (a «Æ i ), B (b «Æ i ) It follows from (51) that A, B 1 So the Kähler angle (defined in [5]) of f is cos tr(aa B B ) 1
148 J FEI, X JIAO, L XIAO AND X XU The structure equations of S with respect to the induced metric can be written as (51) (513) d, d K Æ, where is the complex connection form and K is the Gaussian curvature of S, with respect to the induced metric f ds Using the Maurer Cartan structure equations (59) we obtain d d 1 Æ ( Æ 1 Æ 1 ) It gives Æ 1 Æ 1 by (51) Making use of (59) again, we get d d( Æ 1 Æ 1 ) (1) Æ, which implies K = 1 by (513) Taking the exterior derivative of (511) and using (59), we get the covariant differential of a «Æ i and b «Æ i (defined in [6]) given as follows Da «Æ i p «Æ i q «Æ i Æ, Db «Æ i q «Æ i r «Æ i Æ, where (p «Æ i ) 3 3, (q «Æ i ), (r «i Æ) The second identities q «Æ i imply that f is a minimal immersion ([6]) By the identity (1 15) in [15], the square length of the second fundamental form is (514) S 4 «,i (q «Æ i r «Æ i ) 6 Through some similar straightforward computations, we get the following theorem Theorem 51 The differential geometric quantities of homogeneous -spheres in G(, 4) are given in Table 1, where t ¾ [, ] and c 1 d c d 1, and K is (induced) Gaussian curvature, is the Kähler angle and S is the square length of the second fundamental form REMARK 1 In the case (IV), when 1, f (c 1, c, d 1, d ) are totally geodesic with K They are all U(4)-equivalent to (IV ) f Ï S G(, 4), gt a b b Æ Æa The others in the case (IV) are non-minimal The one given in (III) (resp (III )) with t 4 is U(4)-equivalent to the one given in (IV1) (resp (IV1 )) ([7])
HOMOGENEOUS -SPHERES IN COMPLEX GRASSMANNIANS 149 Table 1 Case Minimality K cos S (I1) Yes 1 1 6 (I1 ) Yes 1 1 6 (I) Yes 5 15 (I ) Yes 5 15 (I3) Yes 3 83 (I3 ) Yes 3 83 (II1) Yes 4 1 (II1 ) Yes 4 1 (III1) Yes 1 (III1 ) Yes 1 (III) Yes 1 4 cos t (III ) Yes 1 4 cos t (IV1) Yes 1 (IV1 ) Yes 1 (IV) 4(1 ) REMARK Since every closed totally geodesic submanifold of a homogeneous Riemannian manifold is homogeneous ([1]), Theorem 51 also contains a complete classification of totally geodesic -spheres in G(, 4) By Table 1 and the above remarks, we obtain Theorem 5 Up to U(4)-equivalent, the one given in (I), (I ), (II1), (II1 ), (III1), (III1 ), (IV1), (IV1 ) and the one given in (IV ) are all totally geodesic -spheres in G(, 4) REMARK 3 The 1-parameter family of homogeneous holomorphic -spheres in (III1) was first discovered by Q-S Chi and Y-B Zheng in [7] REMARK 4 There are some differences between our classification and the classification of minimal -spheres with constant curvature in G(, 4) by Z-Q Li and Z-H Yu ([14]) The case (IV) in our classification is not contained in theirs, and there is a holomorphic (thus minimal) -sphere with constant curvature K 43 in G(, 4) given in [14] which is not contained in ours To conclude this section, we want to prove that the holomorphic -sphere with K 43 in G(, 4) mentioned in Remark 4 is not homogeneous
15 J FEI, X JIAO, L XIAO AND X XU Set S {} and z is the local coordinate of S, and define the map ³ Ï S G(, 4), z 1 3z 1 ¾ Ö 8 3 z Ö 1 3 z Theorem 53 The holomorphic embedding ³ is not homogeneous Proof We choose a unitary frame field e (e 1, e, e 3, e 4 ) along ³ as follows: e 1 1 (1,, 3z, ), 1 e 1 Ö 8 Æzz, 1 3z 4, Ö e 3, 1 1 3 3 Æz,, 1, e 4 1 3Æz 1 13 z, 4 3 z, Ö 1 3 z(1 3z4 ) Ö 8 3 Æz, 1 1 3 z, 3 z,, where 1 1 3z 4, 1 3z 6z 4 1z 6 9z 8 3z 1, 3 4 Ö Ö 1 1 3 z, 1 1 3 z 4z 4 z 6 1 3 z8 By direct computation, we get 1 Æ 3 (de 1, e 3 ), 14 Æ (de 1, e 4 ) 3z(1 (13)z ) dz, 1 4 3(1 3z 3 Æ (de, e 3 ) 4 ) dz, 3 3 Æ 4 (de, e 4 ) 6(z 6 3z 4 1) 3 4 dz
HOMOGENEOUS -SPHERES IN COMPLEX GRASSMANNIANS 151 Thus, the metric induced by ³ is ³ ds 1 Æ 3 Æ 1 Æ 3 1 Æ 4 Æ 1 Æ 4 Æ 3 Æ Æ 3 Æ 4 Æ Æ 4 3 dz dæz, (1 z ) which implies the (induced) Gaussian curvature K 43 Set 3(1 z ) dz, then we have (515) A (1 3z 4 )(1 z ) 3 3 z(1 (13)z )(1 z ) 1 4 (z6 3z 4 1)(1 z ) 3 4 by (511) Up to now, we have two ways to show that ³ is not homogeneous The first one is that z is an isolate zero point of det A and rank A z 1 by (515), which implies that ³ is not homogeneous The second one is that the square length of the second fundamental form of ³ is given by S 16(3 z 3z 4 ) 9(1 z ), by the first identity of (514), which is not a constant, and hence ³ is not homogeneous ACKNOWLEDGMENTS The authors thank the referee for the very helpful remarks and valuable suggestions, which improve the presentation of this paper and make it readable The author Fei would like to thank Prof Y Ohnita for kindly private discussion and some constructive suggestions Fei and Xu also would like to express appreciation to Prof Jiagui Peng for his helpful guidance References [1] S Bando and Y Ohnita: Minimal -spheres with constant curvature in P n (C), J Math Soc Japan 39 (1987), no 3, 477 487 [] J Bolton, GR Jensen, M Rigoli and LM Woodward: On conformal minimal immersions of S into CP n, Math Ann 79 (1988), 599 6 [3] RL Bryant: Minimal surfaces of constant curvature in S n, Trans Amer Math Soc 9 (1985), 59 71 [4] E Calabi: Minimal immersions of surfaces in Euclidean spheres, J Differential Geometry 1 (1967), 111 15 [5] SS Chern and JG Wolfson: Minimal surfaces by moving frames, Amer J Math 15 (1983), 59 83
15 J FEI, X JIAO, L XIAO AND X XU [6] SS Chern and JG Wolfson: Harmonic maps of the two-sphere into a complex Grassmann manifold, II, Ann of Math () 15 (1987), 31 335 [7] Q-S Chi and Y Zheng: Rigidity of pseudo-holomorphic curves of constant curvature in Grassmann manifolds, Trans Amer Math Soc 313 (1989), 393 46 [8] MP do Carmo and NR Wallach: Representations of compact groups and minimal immersions into spheres, J Differential Geometry 4 (197), 91 14 [9] MP do Carmo and NR Wallach: Minimal immersions of spheres into spheres, Ann of Math () 93 (1971), 43 6 [1] X Jiao and J Peng: Classification of holomorphic spheres of constant curvature in complex Grassmann manifold G,5, Differential Geom Appl (4), 67 77 [11] K Kenmotsu: On minimal immersions of R into S N, J Math Soc Japan 8 (1976), 18 191 [1] S Kobayashi and K Nomizu: Foundations of Differential Geometry, II, Interscience Publishers John Wiley & Sons, Inc, New York, 1969 [13] H Li, C Wang and F Wu: The classification of homogeneous -spheres in P n, Asian J Math 5 (1), 93 18 [14] Z-Q Li and Z-H Yu: Constant curved minimal -spheres in G(, 4), Manuscripta Math 1 (1999), 35 316 [15] K Ogiue: Differential geometry of Käehler submanifolds, Advances in Math 13 (1974), 73 114 Jie Fei Mathematics and Physics Center Xi an Jiaotong-Liverpool University Suzhou 1513 PR China e-mail: feijie311@163com Xiaoxiang Jiao Department of Mathematics Graduate University, CAS Beijing 149 PR China e-mail: xxjiao@gucasaccn Liang Xiao Department of Mathematics Graduate University, CAS Beijing 149 PR China e-mail: lxiao@gucasaccn Xiaowei Xu Department of Mathematics USTC Hefei 36 PR China e-mail: xwxu9@ustceducn